# Is the Hankel Transform a Hankel Operator

The "Hankel Transform" is the infinite weighted sum of the Bessel function. At the top of the wikipedia article http://en.wikipedia.org/wiki/Hankel_transform it says

Not to be confused with the determinant of the Hankel matrix of a sequence

But in integral form, the Hankel transform is given by

$$\int_0^\infty f(r) J_v (kr) r dr$$

(according to the same wikipedia page). On the other hand, the operator form of a Hankel matrix (the one wikipedia says is unrelated) is given by

$$\Gamma x(t)=\int_0^\infty h(t+s) x(s) ds$$

on $L_2(0,\infty)$ . These two forms look suspiciously similar; is the Hankel transform in fact a Hankel operator?

I don't see any reference that claims that they are, e.g., Bowman's "An introduction to Bessel Functions" or Spanier's "An Atlas of Functions" both describe Hankel transforms in considerate detail without mentioning their representation as a Hankel operator.

Note 1: my second definition (of Gamma) comes from "An Introduction to Hankel Operators" by Partington, which I don't think is online, but https://www.encyclopediaofmath.org/index.php/Hankel_operator seems to gives a similar, if more confusingly formulated version of the definition.

Note 2: I understand that the distinction wikipedia is making is that the two functions referred to as Hankel transforms are not the same, in that regardless if the Bessel-Hankel transform is a Hankel operator, it isn't the determinant of a Hankel matrix.

Note 3: I've tagged with "Matrix" because Hankel matrices are very common, and someone understanding Hankel matrices might have a good answer to this question if this is incorrect please remove tag